Nambu–Goldstone Bosons

The existence of gapless excitations, the Nambu–Goldstone bosons, is the most striking consequence of spontaneous symmetry breaking. This chapter starts with an intuitive explanation of the presence of such gapless modes in the spectrum. The insight thus developed is already sufficient to understand the intimate relationship between the number of Nambu–Goldstone modes, their dispersion relations, and the nature of broken symmetry. The argument is then put on a solid mathematical basis. First, the Goldstone theorem, which asserts the very existence of Nambu–Goldstone bosons, is proven. The text then continues with a discussion of redundancies between certain, in particular spacetime, symmetries. This leads to a prediction for the number of independent Nambu–Goldstone fields. Finally, a counting rule relating the numbers of independent fields and states in the spectrum is established. The chapter concludes with a brief overview of Nambu–Goldstone-like modes, which are also related to symmetry yet lack one or more of the attributes of genuine Nambu–Goldstone bosons.